高斯和与有理点的计算

计算有限域上代数簇有理点个数是有限域研究中的重要课题. 设为q元有限域, f是上的非零多项式, D_f为其次数矩阵, 用N(f)表示超曲面f=0在上的有理点个数. 若D_f在剩余类环中与整数矩阵A行等价, 则记为D_f ~_qA. 利用高斯和给出了当D_f ~_q diag(), 其中∈{1, p₁}, p₁为q-1的一个素因子时N(f) 的具体表达式, 从而推广了已知的结论.

Gauss sums and computation of rational points

It is important in the area of finite fields study to compute the number of rational points on algebraic varieties in finite fields. Let be the finite field of order q and f be a nonzero polynomial over with degree matrix D_f; denote by N(f) the number of -rational points of the hypersurface defined by f=0; write D_f ~_qA when D_f is row equivalent to an integer matrix A in the residue ring . Using Gauss sums, we obtain the explicit formula for N(f) in the case of D_f ~_q diag(), where and p₁ is a prime divisor of q-1, which generalizes the known results.